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oai:recercat.cat:2445/965942025-12-05T09:56:40Zcom_2072_1057col_2072_478917col_2072_478920

A Hilbert-Mumford criterion for polystability in Kahler geometry Mundet i Riera, Ignasi Geometria algebraica Geometria Algebraic geometry Geometry Consider a Hamiltonian action of a compact Lie group $ K$ on a Kaehler manifold $ X$ with moment map $ \mu:X\to\mathfrak{k}^*$. Assume that the action of $ K$ extends to a holomorphic action of the complexification $ G$ of $ K$. We characterize which $ G$-orbits in $ X$ intersect $ \mu^{-1}(0)$ in terms of the maximal weights $ \lim_{t\to\infty}\langle\mu(e^{\mathbf{i} ts}\cdot x),s\rangle$, where $ s\in\mathfrak{k}$. We do not impose any a priori restriction on the stabilizer of $ x$. Under some mild restrictions on the action $ K\circlearrowright X$, we view the maximal weights as defining a collection of maps: for each $ x\in X$, $\displaystyle \lambda_x:\partial_{\infty}(K\backslash G)\to\mathbb{R}\cup\{\infty\},$ where $ \partial_{\infty}(K\backslash G)$ is the boundary at infinity of the symmetric space $ K\backslash G$. We prove that $ G\cdot x\cap\mu^{-1}(0)\neq\emptyset$ if: (1) $ \lambda_x$ is everywhere nonnegative, (2) any boundary point $ y$ such that $ \lambda_x(y)=0$ can be connected with a geodesic in $ K\backslash G$ to another boundary point $ y'$ satisfying $ \lambda_x(y')=0$. We also prove that the maximal weight functions are $ G$-equivariant: for any $ g\in G$ and any $ y\in \partial_{\infty}(K\backslash G)$ we have $ \lambda_{g\cdot x}(y)=\lambda_x(y\cdot g)$. 2016-03-17T16:42:48Z 2016-03-17T16:42:48Z 2010 2016-03-17T16:42:53Z info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion 0002-9947 https://hdl.handle.net/2445/96594 609881 eng Reproducció del document publicat a: http://dx.doi.org/10.1090/S0002-9947-2010-04831-7 Transactions of the American Mathematical Society, 2010, vol. 362, p. 5169-5187 http://dx.doi.org/10.1090/S0002-9947-2010-04831-7 (c) American Mathematical Society (AMS), 2010 info:eu-repo/semantics/openAccess 19 p. application/pdf American Mathematical Society (AMS) Articles publicats en revistes (Matemàtiques i Informàtica)